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Require Export base.

Section preorder.
  Context `{SubsetEq A} `{!PreOrder ()}.

  Global Instance preorder_equiv: Equiv A := λ X Y, X  Y  Y  X.
  Instance preorder_equivalence: @Equivalence A ().
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  Proof.
    split.
    * firstorder.
    * firstorder.
    * intros x y z; split; transitivity y; firstorder.
  Qed.
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  Global Instance: Proper (() ==> () ==> iff) ().
  Proof.
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    unfold equiv, preorder_equiv.
    intros x1 y1 ? x2 y2 ?. split; intro.
    * transitivity x1. tauto. transitivity x2; tauto.
    * transitivity y1. tauto. transitivity y2; tauto.
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  Qed.
End preorder.

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Hint Extern 0 (@Equivalence _ ()) =>
  class_apply preorder_equivalence : typeclass_instances.
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Section bounded_join_sl.
  Context `{BoundedJoinSemiLattice A}.

  Hint Resolve subseteq_empty subseteq_union_l subseteq_union_r union_least.

  Lemma union_compat_l x1 x2 y : x1  x2  x1  x2  y.
  Proof. intros. transitivity x2; auto. Qed.
  Lemma union_compat_r x1 x2 y : x1  x2  x1  y  x2.
  Proof. intros. transitivity x2; auto. Qed.
  Hint Resolve union_compat_l union_compat_r.

  Lemma union_compat x1 x2 y1 y2 : x1  x2  y1  y2  x1  y1  x2  y2.
  Proof. auto. Qed.
  Lemma union_empty x : x    x.
  Proof. apply union_least. easy. auto. Qed.
  Lemma union_comm x y : x  y  y  x.
  Proof. auto. Qed.
  Lemma union_assoc_1 x y z : (x  y)  z  x  (y  z).
  Proof. auto. Qed.
  Lemma union_assoc_2 x y z : x  (y  z)  (x  y)  z.
  Proof. auto. Qed.

  Global Instance: Proper (() ==> () ==> ()) ().
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  Proof.
    unfold equiv, preorder_equiv. split; apply union_compat; simpl in *; tauto.
  Qed.
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  Global Instance: Idempotent () ().
  Proof. split; eauto. Qed.
  Global Instance: LeftId ()  ().
  Proof. split; eauto. Qed.
  Global Instance: RightId ()  ().
  Proof. split; eauto. Qed.
  Global Instance: Commutative () ().
  Proof. split; apply union_comm. Qed.
  Global Instance: Associative () ().
  Proof. split. apply union_assoc_2. apply union_assoc_1. Qed.

  Lemma subseteq_union X Y : X  Y  X  Y  Y.
  Proof. repeat split; eauto. intros E. rewrite <-E. auto. Qed.
  Lemma subseteq_union_1 X Y : X  Y  X  Y  Y.
  Proof. apply subseteq_union. Qed.
  Lemma subseteq_union_2 X Y : X  Y  Y  X  Y.
  Proof. apply subseteq_union. Qed.

  Lemma equiv_empty X : X    X  .
  Proof. split; eauto. Qed.
End bounded_join_sl.

Section meet_sl.
  Context `{MeetSemiLattice A}.

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  Hint Resolve subseteq_intersection_l subseteq_intersection_r
    intersection_greatest.
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  Lemma intersection_compat_l x1 x2 y : x1  x2  x1  y  x2.
  Proof. intros. transitivity x1; auto. Qed.
  Lemma intersection_compat_r x1 x2 y : x1  x2  y  x1  x2.
  Proof. intros. transitivity x1; auto. Qed.
  Hint Resolve intersection_compat_l intersection_compat_r.

  Lemma intersection_compat x1 x2 y1 y2 : x1  x2  y1  y2  x1  y1  x2  y2.
  Proof. auto. Qed.
  Lemma intersection_comm x y : x  y  y  x.
  Proof. auto. Qed.
  Lemma intersection_assoc_1 x y z : (x  y)  z  x  (y  z).
  Proof. auto. Qed.
  Lemma intersection_assoc_2 x y z : x  (y  z)  (x  y)  z.
  Proof. auto. Qed.

  Global Instance: Proper (() ==> () ==> ()) ().
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  Proof.
    unfold equiv, preorder_equiv. split;
      apply intersection_compat; simpl in *; tauto.
  Qed.
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  Global Instance: Idempotent () ().
  Proof. split; eauto. Qed.
  Global Instance: Commutative () ().
  Proof. split; apply intersection_comm. Qed.
  Global Instance: Associative () ().
  Proof. split. apply intersection_assoc_2. apply intersection_assoc_1. Qed.

  Lemma subseteq_intersection X Y : X  Y  X  Y  X.
  Proof. repeat split; eauto. intros E. rewrite <-E. auto. Qed.
  Lemma subseteq_intersection_1 X Y : X  Y  X  Y  X.
  Proof. apply subseteq_intersection. Qed.
  Lemma subseteq_intersection_2 X Y : X  Y  X  X  Y.
  Proof. apply subseteq_intersection. Qed.
End meet_sl.